The answer to your question would be that the rule that best explains why the verb in the following sentence is correct is the following one: This sentence has a mixed compound subject connected by nor, so the verb agrees with the singular subject that is closer.
In this case, "neither" is part of a two-part conjunction joining two subjects: "the council members" and "the mayor". Therefore, the verb must agree with the nearer subject ("the mayor", which is singular).
